The Zak transform and some counterexamples in time-frequency analysis

It is shown how the Zak transform can be used to find nontrivial examples of functions f, g in L/sup 2/(R) with f*g identical to 0 identical to F*G, where F, G are the Fourier transforms of f, g, respectively. This is then used to exhibit a nontrivial pair of functions h, k in L/sup 2/(R), h not=k, such that mod h mod = mod k mod , mod H mod = mod K mod . A similar construction is used to find an abundance of nontrivial pairs of functions h, k in L/sup 2/(R), h not=k, with mod A/sub h/ mod = mod A/sub k/ mod or with mod W/sub h/ mod = mod W/sub k/ mod where A/sub h/, A/sub k/ and W/sub h/, W/sub k/ are the ambiguity functions and Wigner distributions of h, k, respectively. One of the examples of a pair of h, k in L/sup 2/(R), h not=k, with mod A/sub h/ mod = mod A/sub k/ mod is F.A. Grunbaum's (1981) example. In addition, nontrivial examples of functions g and signals f/sub 1/ not=f/sub 2/ such that f/sub 1/ and f/sub 2 /have the same spectrogram when using g as window have been found.<
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